Question

(i) lim
log(3+x) -log(3- x)
x 0
х​

Answer 1

To Find:-

lim X -> 0 [ log(3 + x) - log(3 - x) ] ....(corrected Question)

Solution:-

lim x-> 0 log(3 + x) - log(3 - x)

lim x -> 0 log((3 + x)/(3 - x))

lim x -> 0 log[((3(1 + x/3))/(3(1 - x/3))]

lim x -> 0 log[( 1 + x/3)/ ( 1 - x/3)]

• Dividing numerator and denominator by x/3

•°• lim x -> 0 log [(1 + x/3)/(x/3) / (1 - x/3)/(x/3)]

•°• lim x-> 0 log (1 + x/3)/(x/3) - log(1 - x/3)/(x/3)

_____________________________

By limit of logarithm

=> lim x ->0 log (1 + x)/x = 1

________________________________

•°• lim x->0 log (1 + x/3)/(x/3) - lim x ->0 log(1 - x/3)/(x/3)

•°• = 1 - (-1) = 1 + 1 = 2

• Answer

lim X -> 0 [ log(3 + x) - log(3 - x) ] = 2